Families of Riemann surfaces and Weil-Petersson geometry

TL;DR

This paper provides an overview of Weil-Petersson geometry of Riemann surfaces, discussing recent developments, key research results, and outlining future lecture plans, excluding Mirzakhani's Witten-Kontsevich work.

Contribution

It summarizes significant advances in WP geometry since 2008, including ergodicity, geodesic behavior, and curvature properties, and sets the stage for further educational lectures.

Findings

WP geodesic flow is ergodic

Asymptotics of WP geodesics analyzed

Geodesic-length functions and curvature tensor studied

Abstract

A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich were given at the 2011 Park City Mathematics Institute. Research on WP geometry coming after 2008 includes the Brock-Masur-Minsky second paper, Asymptotics of WP geodesics II: bounded geometry and unbounded entropy; the Burns-Masur-Wilkinson, The WP geodesic flow is ergodic; and the Wolpert, Geodesic-Length Functions and the WP curvature tensor.